Elie Assémat

Quantum Dynamics in Phase Space using Projected von Neumann Bases

We describe the mathematical underpinnings of the biorthogonal von Neumann method for quantum mechanical simulations (PvB). In particular, we present a detailed discussion of the important issue of nonorthogonal projection onto subspaces of biorthogonal bases, and how this differs from orthogonal projection. We present various representations of the Schrödinger equation in the reduced basis and discuss their relative merits. We conclude with illustrative examples and a discussion of the outlook and challenges ahead for the PvB representation.

Gradient optimization of analytic controls: the route to high accuracy quantum optimal control

We argue that quantum optimal control can and should be done with analytic control functions, in the vast majority of applications. First, we show that discretizing continuous control functions as piecewise-constant functions prevents high accuracy optimization at reasonable computational costs. Second, we argue that the number of control parameters required is on-par with the dimension of the object manipulated, and therefore one may choose parametrization by other considerations, e.g. experimental suitability and the potential for physical insight into the optimized pulse. Third, we note that optimal control algorithms which make use of the gradient of the goal function with respect to control parameters are generally faster and reach higher final accuracies than non gradient-based methods. Thus, if the gradient can be efficiently computed, it should be used. Fourth, we present a novel way of computing the gradient based on an equation of motion for the gradient, which we evolve in time by the Taylor expansion of the propagator. This allows one to calculate any physically relevant analytic controls to arbitrarily high precision. The combination of the above techniques is GOAT (Gradient Optimization of Analytic conTrols) gradient-based optimal control for analytic control functions, utilizing exact evolution in time of the derivative of the propagator with respect to arbitrary control parameters.Quantum computation places very stringent demands on gate fidelities, and experimental implementations require both the controls and the resultant dynamics to conform to hardware-specific constraints. Superconducting qubits present the additional requirement that pulses must have simple parameterizations, so they can be further calibrated in the experiment, to compensate for uncertainties in system parameters. Other quantum technologies, such as sensing, require extremely high fidelities. We present a novel, conceptually simple and easy-to-implement gradient-based optimal control technique named gradient optimization of analytic controls (GOAT), which satisfies all the above requirements, unlike previous approaches. To demonstrate GOATs capabilities, with emphasis on flexibility and ease of subsequent calibration, we optimize fast coherence-limited pulses for two leading superconducting qubits architectures-flux-tunable transmons and fixed-frequency transmons with tunable couplers.

On the control by electromagnetic fields of quantum systems with infinite dimensional Hilbert space

We analyze the control by electromagnetic fields of quantum systems with infinite dimensional Hilbert space and a discrete spectrum. Based on recent mathematical results, we rigorously show under which conditions such a system can be approximated in a finite dimensional Hilbert space. For a given threshold error, we estimate this finite dimension in terms of the used control field. As illustrative examples, we consider the cases of a rigid rotor and of a harmonic oscillator.